Lines and Planes in Space

Vectors let us describe lines and planes in any dimension by starting from a fixed point and moving along directions. A line through a point with position vector $r_0$ and direction vector $d$ is defined by:

$r = r_0 + t d, \quad t \in \mathbb{R}$

The parameter $t$ slides you along the line, while $d$ sets the heading. Unlike slope-intercept form, this parametric form works identically in 2D, 3D, and beyond.

A plane is captured by a normal vector $n$ that is perpendicular to the plane. A point $r$ lies in the plane through $r_0$ exactly when the displacement is orthogonal to $n$:

$n \cdot (r - r_0) = 0$

Expanded in coordinates, this yields $ax + by + cz = d$, where $(a, b, c)$ is the normal vector.

Vector Relations

Translating geometry into vector algebra makes spatial tests routine. The relations between planes depend directly on their normal vectors:

Common Pitfall: Never confuse a plane's normal vector with a direction lying in the plane. The normal is perpendicular to the surface and provides the coefficients $(a, b, c)$. Using an in-plane direction instead of the normal completely inverts the geometry.